ON THE CONTINUOUS FINITE-TIME STABILIZATION OF THE DOUBLE INTEGRATOR

Continuous finite-time stabilization is often treated under the analytical framework of homogeneity and has been frequently illustrated in the context of the feedback control of the double integrator. For such a simple system, the simplest considered continuous finite-time controller is composed of gained (proportional) exponentially weighted position and velocity error correction terms, with the exponential weights generally less than unity and constrained to satisfy a particular relation among them under homogeneity. What happens for less-than-unity exponential weights that do not satisfy such a homogeneity-based relation? Does the finite-time stabilization hold? Through a Lyapunov function--based study, we analyze and give more concrete answers to such questions than those partially provided by previous studies on the topic. We do find a more exhaustive spectrum of the exponential weights that give rise to finite-time stability of the trivial solution. Other types of stability properties are further found to take place for less-than-or-equal-to-unity exponential weights. Moreover, through complementary analysis, local or ultimate behavior of the system solutions is further characterized. The analytical findings are further illustrated through computer simulations.

Automated summary

This paper focuses on continuous finite-time stabilization under the framework of homogeneity, analyzing the effects of exponential weights less than unity that do not satisfy homogeneity-based relation. A broader spectrum of exponential weights leading to finite-time stability is identified, along with other stability properties for weights less than or equal to unity.